Problem 26 - Algebra Form and Function

Question

Suppose two samples of 5 values are taken from a population $$ a: 8,9,4,7,5 \text { and } b: 6,9,10,5,7 $$ (a) Find \(\bar{a}\) and \(\bar{b}\). (b) Find the mean of the sample you get by combining the two samples. (c) Is the mean of the combined sample equal to the mean of the two values \(\bar{a}\) and \(\bar{b}\) ? (d) Explain your answer in (c) algebraically.

Step-by-Step Solution

Answer

Question: Explain the relationship between the means of two individual samples and the mean of their combined sample. Answer: The mean of the combined sample is equal to the average of the means of the individual samples. Algebraically, this can be represented as \(\bar{ab} = \frac{\bar{a} + \bar{b}}{2}\). This relationship holds true because the combined sample mean is essentially the sum of the values of the individual samples divided by the total number of values, which is the same as taking the average of the individual means.

Step 1: a) Find the mean of sample a (\(\bar{a}\)) and b (\(\bar{b}\)).

To find the mean of samples a and b, we will sum the values for each sample and divide by the number of values in the sample which 5 in either case. For a: $$ \bar{a} = \frac{8 + 9 + 4 + 7 + 5}{5} $$ For b: $$ \bar{b} = \frac{6 + 9 + 10 + 5 + 7}{5} $$

Step 2: b) Calculate the mean of the combined sample.

Now, we will find the mean of the combined sample by summing up all the values of the two samples (a and b) and then dividing by 10, as there are a total of 10 values in the combined sample. Mean of Combined sample: $$ \bar{ab} = \frac{8 + 9 + 4 + 7 + 5 + 6 + 9 + 10 + 5 + 7}{10} $$

Step 3: c) Compare the mean of the combined sample with the average of the individual means (\(\bar{a}\) and \(\bar{b}\)).

In order to find out if the mean of the combined sample (\(\bar{ab}\)) is equal to the mean of the two values of the individual means (\(\bar{a}\) and \(\bar{b}\)), we will: Find the average of the individual means and compare it to the calculated mean of the combined sample. $$ \frac{\bar{a} + \bar{b}}{2} $$

Step 4: d) Explain your answer in (c) algebraically.

Now we will explain the relation between the means algebraically, let's denote the sum of a as: $$ S_a = a_1 + a_2 + a_3 + a_4 + a_5 $$ And the sum of b as: $$ S_b = b_1 + b_2 + b_3 + b_4 + b_5 $$ We can represent the individual means in terms of the sums, $$ \bar{a} = \frac{S_a}{5}, \quad \bar{b} = \frac{S_b}{5} $$ Similarly, the mean of the combined sample is, $$ \bar{ab} = \frac{S_a + S_b}{10} $$ Now let's find the average of the individual means, $$ \frac{\bar{a} + \bar{b}}{2} = \frac{\frac{S_a}{5} + \frac{S_b}{5}}{2} = \frac{S_a + S_b}{10} $$ Comparing the mean of the combined sample (\(\bar{ab}\)) to the average of the individual means (\(\bar{a}\) and \(\bar{b}\)), we can see that both expressions are equal: $$ \bar{ab} = \frac{\bar{a} + \bar{b}}{2} = \frac{S_a + S_b}{10} $$

Key Concepts

Sample Mean

The sample mean is a fundamental concept in statistics used to measure the central tendency of a set of data. It helps us summarize a data set with a single value that represents its average. When we look at two different samples, like the ones in this problem, finding the sample mean for each can provide insightful statistical summaries of the data.
To find the sample mean for sample 'a', you simply sum up all the values in the sample and then divide this sum by the number of values in the sample. Let's break it down:

• Add up all the values: 8, 9, 4, 7, and 5, giving us a total of 33.
• Divide this sum by the number of values (5) to find \(\bar{a} = \frac{33}{5} = 6.6\).

Similarly, for sample 'b':

• Add up 6, 9, 10, 5, and 7 to get a total of 37.
• Then divide by 5, which results in \(\bar{b} = \frac{37}{5} = 7.4\).

These means give us a clearer view of how each sample behaves, allowing us to compare individual datasets quantitatively.

Combined Sample Mean

The combined sample mean is an average that is calculated when two or more samples are merged into a single data set. It helps in understanding the overall trend across multiple data sets. To calculate the combined mean, we first combine all the data points from each sample into a single set and then find the mean of this collective set.
In this problem, the combined sample includes all data points from both samples 'a' and 'b': namely 8, 9, 4, 7, 5, 6, 9, 10, 5, and 7. Here's how you calculate it:

• First, sum up all these values which gives you a total of 70.
• Then, divide this total by the number of combined points (10) to get \(\bar{ab} = \frac{70}{10} = 7.0\).

The combined sample mean provides a single value that represents the mean of the entire data set, allowing for an easy comparison to an average of individual sample means.

Algebraic Explanation

Understanding the relation between the means of individual samples and the combined sample is enhanced through algebraic manipulation. When comparing the combined sample mean to the average of the individual sample means, we see that both yield the same result due to the simplifications in mathematics.
Let's denote:

• \(S_a\) and \(S_b\) as the sums of samples 'a' and 'b', respectively.
• \(\bar{a} = \frac{S_a}{5}\) and \(\bar{b} = \frac{S_b}{5}\) represent the individual sample means.

For the combined sample mean, you add the sums of both samples and divide by the total number of data points. Mathematically:

• \(\bar{ab} = \frac{S_a + S_b}{10} = 7.0\).

Now, the average of the individual means would be:

• \(\frac{\bar{a} + \bar{b}}{2} = \frac{\frac{S_a}{5} + \frac{S_b}{5}}{2} = \frac{S_a + S_b}{10} = 7.0\).

Hence, algebraically both calculations lead to the same value, affirming that the mean of the combined sample is equal to the average of the individual means.

Statistical Analysis

Statistical analysis is a powerful tool that allows us to interpret data and make informed decisions. By using means, both individual and combined, we gain insight into the data set’s general trends and behavioral characteristics. This analysis not only involves understanding how data can be merged but also the impacts on measures of central tendency like mean.
Understanding the mean at both an individual sample level and a combined sample level helps in:

• Identifying differences between subsets of data.
• Understanding how combining data samples affects overall data interpretation.
• Comparing the spread or variability within and between data sets.

By recognizing these elements, students can discern how data should be manipulated and interpreted in larger contexts. It’s about comprehending how means transform when datasets merge, impacting forecasting or decision-making based on data-driven insights.